Annexes in affine Coxeter complexes

Abstract

We introduce the annex of an element $x$ in a Coxeter group as the set of elements $y$ such that $x \nleq y$ with respect to Bruhat order. This notion provides a complementary perspective to the study of Bruhat intervals and their interpretation via folded galleries. We establish general properties of annexes and show that in affine Coxeter groups the annex of any fixed element is finite. In rank-two affine Coxeter complexes, we further describe the geometric structure of annex boundaries using descent sets and configurations of parallel reflections. These results offer a new geometric viewpoint on the structure of the Bruhat order.

Figures (6)

• Figure 1: Two galleries in $\tilde{A}_2$. The grey gallery is unfolded, and the black gallery is folded at two panels.
• Figure 2: Here we have a gallery shown in black. The grey line shows the corresponding unfolded gallery and the dashed line shows its footprint.
• Figure 3: The alcoves shaded yellow are in the preshadow of $x=s_0s_1s_2s_0s_1s_0s_2$.
• Figure 4: The annex of $x=s_0s_1s_2s_0s_1s_0s_2$ is the set of alcoves shaded yellow.
• Figure 5: In (a), we have the annex of the alcove $w=s_0s_2s_1s_0s_2s_0$, with $D_R(w)=\{1\}$. In (b), we have the annex of the alcove $ws_1$. We can obtain picture (b) from picture (a) by expanding the set to include every element of the form $y W_{\{0,2\}}$, where $y$ is a shaded alcove in (a). This will include all alcoves from (a), shown in (b) as the highlighted section.

Theorems & Definitions (146)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Example 2.2
• Example 2.3
• Proposition 2.4
• Definition 2.5
• Example 2.6